Hollow Sphere Formula Derivation

The Moment of Inertia is a quantity that defines the torque required for a preferred rotational motion around a rotational axis. It relies on the mass distribution of the object and the axis chosen, with bigger moments requiring more torque to affect the rate of rotation. A hollow sphere is a sphere that has been thinned out to the point that a wall of equivalent thickness produces an interior ball inside the exterior ball. Let’s discuss the moment of inertia of a hollow sphere. 

Moment of Inertia

The moment of inertia of a rigid body, also known as the mass moment of inertia,  is a measure that defines the torque required for a preferred angular acceleration about an axis of rotation, in the same way that mass defines the force required for a preferred acceleration. It relies on the mass distribution of the object and the axis selected, with higher moments necessitating more torque to affect the rate of spin.

The volume of a hollow sphere

A 3D figure’s volume is described as the figure’s capacity, or the quantity of content it can store. The volume of a hollow sphere is calculated by subtracting the volume of the interior sphere from the volume of the exterior sphere. It may be computed using the formula: Volume of the sphere is:

= 4/3 πR³

Thus volume of hollow sphere= volume of exterior (outer) sphere – volume of interior (inner) sphere

= 4/3 πR³-4/3 πr³

= 4/3 π(R³-r³)

The curved surface area of the hollow sphere: 

The area of the sheet that may fully lie on top of the hollow sphere is known as the curved surface area. It equals the Curved Surface Area of the interior sphere minus the Curved Surface Area of the outer sphere.

The curved Surface area of a hollow sphere,

= Curved Surface Area of the outer sphere – Curved Surface Area of the inner sphere

= 4 πR² – 4 πr²

= 4 π(R²-r²) 

The total surface area of a hollow sphere is the same as the Curved Surface Area of a hollow sphere since a hollow sphere has just one surface.

For a hollow sphere, Curved Surface Area =Total Surface Area.

Moment of inertia of a hollow sphere

The Moment of Inertia, also known as the mass moment of inertia of a rigid body, is a quantity that defines the torque required for a preferred rotational motion around a rotational axis, much like the mass defines the force needed for the wanted acceleration. This appears to be fully reliant on the mass distribution of the object and the axis is chosen, with bigger moments requiring more torque to affect the object’s rate of rotation.

The following formula has been used to determine the moment of inertia of a hollow sphere:

I = MR²

Derivation of Hollow Sphere Formula

The Moment of Inertia of a Circle is given by,

I = mr²

Application of differential analysis the result is;

dl = r² dm

It is required to find the dm value utilising the formula,

dm = dA

dA = R dθ × 2πr

In where A is the entire surface area of the sphere, which is equal to 4R2 and dA is the area of the circle generated via differentiation, which is equal to;

R dθ is the thickness and 2πr is the circumference of the circle.

We get R dθ from the formula of arc length, S = R θ

 sin θ =  r = R sinθ

Then, dA is:.

dA = 2πR²sinθ dθ

When exchanging the equation for dA into dm,

dm = dθ

Substitute the equation for ‘r’ into the equation for ‘dI.’ 

dm = sin³ θ dθ

So, if we integrate through one end to the other inside the range of 0 to radians, we get:

I = sin³ θ dθ

Then we must divide sin³ into two parts, as it represents the integral of odd power trigonometric functions:

I = sin² θ sin θ dθ

sin² θ is given as sin² θ = 1- cos² θ.

Thus,

I = (1- cos² θ) sin θ dθ

We can use substitution, where u = cos θ, 

I = u² – 1 du

I = u² – 1 du,

In this, the integral of u² du = u as well as the integral of 1 du = u

When substituting the values, we get.

I = (1-¹ – u1-¹ )

I = {[ (-1)³ -1³] − − 1−1}

I = {– −2}

I = { +2}

I = x

I = MR²

Conclusion

To conclude, the volume of a hollow sphere is calculated by subtracting the interior sphere from the volume of the exterior sphere. The Moment of Inertia, also known as the mass moment of inertia of a rigid body, is a quantity that defines the torque required for a preferred rotational motion around a rotational axis. A hollow sphere is a thinned ball with a wall of similar thickness that generates an interior ball within the exterior ball. While mass is thought of as inertia, each has its moment of inertia. The moment of inertia of a hollow sphere formula is given in the article above.